Furthermore, why not have these factions tie into the character's backgrounds, supplying them at least a bond and an ideal? After all, it would be strange for a character to join a faction whose ideals they didn't (at least pretend to) share. RIGHT?

ANOTHER PROBLEM

The factions as described on WotC's site are pretty tame and lackluster to me. Very safe and uncontroversial and strangely defensive, as if you're worried that the religious Order of the Gauntlet is going to be too, you know, religious.

SO HERE'S WHAT I'VE DONE

I've made several changes and added new content for when you're running your #hdqRemix.

Every PC begins as a faction member; choose or roll d6 for it (on 6, DM chooses)

Every PC takes a bond with the immediate superior mentioned in the faction description.

This can be the only bond or simply add to your existing bond-stable.

For each bond, think of something that you want but haven't been able to get from the bond

Affection

Commitment

Repentance

Apology

Restitution

Vengeance

Attention

Conversion

Forgiveness

Admission

Respect

Support (in some venture, plan, or decision)

Every PC takes an ideal from the faction's list; roll or choose. Ideals can always change later and probably should as the character develops.

I've added a bond-NPC for each faction that is supposed to be the contact for low level operatives like the PCs. If multiple PCs are in the same faction, they all share the same "commanding officer".

I've added ideals for each faction that show you what the faction is like

I've added sample missions for which the faction will reward you; you could hunt cultists and do faction play for an entire campaign. (Hack: if you want to expand and foreground faction play, you could hand out XP for faction missions.)

THE FACTIONS

I've just done these quickly. Feel free to contribute or critique in the comments. I have no Forgotten Realms knowledge at all and am not trying to be faithful to its spirit; this is just me trying to make the info on the WotC website a bit spicier.

ORDER OF THE GAUNTLET (Religious)

Bond

Ideals

Purity. Heretics and apostates must be killed; let not your eye pity them.

Impartiality. No exceptions, ever, for anyone.

Mercy. The one who repents must be shown mercy as I have been shown mercy.

Idealism. Fiat justicia ruat caelum.

Deontology. There is no right and wrong. There is only what the god commands and forbids.

Herem. The cult, their families, livestock, and all their household must be slain, and all their goods given to the god.

Missions

Kill an apostate

Bring back a faction member who killed someone who deserved mercy

Kill a heretic; it is a PC's friend or relative

Bring back a cultist for interrogation and sacrifice

Bring back ears of bandits

Track down a faction member who kept cult wealth for own use

LORD'S ALLIANCE (Political)

Bond

Officer Gunner Sheol; ears to the neck, eyes sunken and clouded; sleepwalking; wants to be saved; never speak first, never elaborate, never leave their eyes.

Ideals

Order. The forms must be observed in all things.

Duty. Dulce et decorum est pro patria mori.

Honor. Never dishonor a rightful superior, regardless of the superior's worthiness.

Justice. No mercy.

Practicality. Idealism is for philosophers and the dead.

Hierarchy. The stratified world is beautiful; change leads to chaos.

Missions

Assassinate a rabblerousing villager

Infiltrate and entrap an antiwar/opposition/rival group

Sneak into the enemy camp and assassinate its leader

Pose as a buyer of illicit components, confiscate and apprehend dealers

Rescue city-state citizens being held captive by rival city

Propagandize on behalf of the lord; avoid or intimidate opposing thugs

EMERALD ENCLAVE (Philosophical)

Bond

The Anthrogogue, Corridor Angelspawn; clothes neat and crisp like a mannequin; skin tattooed with equations; bursting into song; wants to be like the Emeraldian; speak energetically, lapsing into sing-songiness and quotations.

Ideals

Ostentation. The gratuitous display of wealth inspires others to do better.

Respect. No dissing tolerated. If you lose your respect, you lose everything.

Revenge. You will be seen as weak and destroyed if you let a slight to you or your people go unpunished.

Piety. The secret gods will bless those who carry out their wishes.

Equivalence. There is no gift; there is only exchange. Gifts are schemes disguised; never trust the "selfless".

Missions

Silence a talker

Off a pig

Make out with the Antikythera Chalice from the gallery of the Warwick Estate

Attend a diplomatic meeting with another faction; secretly kill one of their member who disrespected the Hisser

The Subtle One has commanded that all the lights in the house of Carnifex be extinguished

Find out why the Zhent's lover gave her a gift and refused repayment; the Zhent's expects an answer soon and will reward greatly.

HARPERS (Egalitarian)

Bond

Citizen Simulacrum Blacktomb; brows prehumanly heavy; massive chest; impatient; giving someone a massage; wants to destroy the house of Lord Ember of Neverwinter for microaggressions suffered at his hands; always smile, speak as to children.

Ideals

Equality. Everyone in bondage must be liberated immediately, regardless of the cost.

Equality. No one must be treated differently. Except those who treat others differently. Not even your family, not even those closest to you. Your loves must become strangers so that all strangers becomes loves.

Equality. Religious bigots must be humiliated and brought into conformity.

Equality. All political systems must enforce equality of economic outcomes.

Equality. All is equal; all is one; make no judgments unless judged.

Equality. Animals must be liberated from the inequality of domestication and consumption.

Missions

Convert a cultist

Capture a cultist for interrogation

Start a riot outside despicable politician's fancy villa

Liberate all the slaves owned by the nearest holder

Set all the prisoners free from the nearest jail

Entrap a firebreathing cleric and blackmail him to stop his hatespeech